Chapter 1

,

Synchronous motor controls, Problems and Modeling ¹

1.1. Introduction

The tremendous importance of rotary synchronous motors in the industrial systems control has been recalled in the general introduction of this book. There is in the professional community a very important emulation to define control structures, simple to materially implant and to design and very efficient ([BOS 86, LEO 90, VAS 90, MIL 89, LEP 90, LAC 94, LAJ 95, GRE 97, LOU 99, STU 00b, LOU 04c, LOU 09]). Nowadays, we can consider that the control structures are based on some very solid basic principles that we will present. Of course, from one designer to another, many alternatives can appear (each manufacturer wants to have their own patents), but we can consider that the basic principles exploited in practice are those that will be covered in Chapters 2 and 3 of this book, which are devoted to the torque controls. The most important key concepts will be: “self-control”, torque control in the “natural” reference frame (often known as the a-b-c reference frame); torque control in the rotor reference frame (often known as “Park reference frame” or d-q reference frame). Indeed, when the torque control is carried out, it is easy to successively implant a speed control, to obtain a device usually called an “electronic speed variator” and then if necessary, a position control. These last questions will be tackled in Chapter 4. This chapter exposes a general modeling of the synchronous motors and is particularly used as introduction to Chapters 2 to 4.

1.2. Problems on the synchronous motor control

1.2.1. The synchronous motor control, a vector control

The synchronous motor has much better performances than the direct current motor ([VAS 90, BEN 07, MUL 06]), but the counterpart has more sophisticated power electronics (an inverter instead of a rectifier or a chopper) and more complex control laws. Indeed, it is necessary to fulfill the “brush-collector” function via the converter control. This requires knowledge of the rotor flux direction. As it is interdependent with the rotor, a mechanical position sensor gives the necessary information. There are applications where we seek not to use a mechanical sensor, but this is the objective of Chapter 8 and Chapter 9. We will see that we must synchronize the currents on the position, which is the “self-control” function (see the note in section 1.4.2.2). Indeed, it is ideally necessary (here we simplify a little) to create a stator field in quadrature with the rotor field: this type of control thus completely deserves the term of “vector control”, a concept that was popularized by the induction motor‘s control [LEO 90, CAR 95, CAN 00, ROB 07]. By this strategy we seek to precisely control those as synchronous motors (and consequently in fine as direct current motors, since – as we will see – there is a very strong analogy between the axis q of the synchronous motor and the armature of the direct current motor).

These vector controls can be conceived by various approaches. Theoretically, the most satisfactory approach uses the Park model (in the rotor reference frame, known as d-q). This is discussed in Chapter 3, but historically it was not the first to be industrially used. An approach in the “natural” model (in the three-phase stator reference frame, known as a-b-c) was initially largely used, especially for the non- salient pole machines, where a “three-single-phase” model (apparently simpler) is usable: this approach leads to a “vector control”, since we always seek to impose a given direction to the stator field. This approach is covered in Chapter 2.

This gives us the chance to recall that control modeling and control design are two different activities. We can regard as “logical” the use of a three-phase model written in the natural reference frame to design an a-b-c control, since there is a similarity between the variables of the model and the variables used to implant the control. In the same way, it is logical to use a Park model written in the d-q reference frame, to design a d-q control, with the same argument. But we can install a three-phase current regulation (thus in a-b-c) and use a d-q model to estimate the performances. This is what we will do in Chapter 2, where signals are alternating, because the variables in the d-q reference frame are “continuous” (constant in steady state). It is easier to estimate the performances of this type of variable, than the performances of the sinusoidal variables in steady state.

Several approaches are thus possible. The approach in the three-phase reference frame has the advantage of respecting the effective magnitudes and therefore preserving the specific functioning of the inverter and supervising the effective current amplitudes (protection and security). This is also a practical approach to consider the non-sinusoidal field distribution machines, in order to preserve the effective form of the induced back electromotive forces (back-EMF)¹. The implementation is also simpler, and can be done with a few programmable components [LOU 99]. It is thus normal that it is used and presented here (the subject of Chapter 2). This “natural” approach has disadvantages: it is more difficult to evaluate its dynamic performances and (correlatively) it is more difficult to have very good performances.

This is why we will devote Chapter 3 to controls design in the d-q reference frame: the implementation is certainly more complex (but really facilitated nowadays by the programmable modern components); on the other hand, the vector control and the controls‘ design imposing the desired dynamics is direct and natural. We can easily obtain very good dynamic performances. Therefore, this approach is frequently used nowadays.

1.2.2. Direct/inverse model and modeling hypotheses

As shown in the previous section, the synchronous motor control is largely based on the mathematical machine model. We also have just seen that the a-b-c model and the d-q model “naturally” lead to controls of different structures (even if it is possible but not necessarily very easy – to go from one to the other). The modeling hypotheses will thus necessarily influence the control algorithms. Indeed, controls, under various names: “input-output linearization with state feedback” (automatics vocabulary), control “with d and q axes decoupling” (electrical engineering vocabulary), “inverse models” control, amount in fact to deducing the model control structure (known as “direct” [HAU 97]) of the motor. The structure, but not necessarily the controller: thus, these approaches show that we must (for example) control the currents (three-phase currents or Park components of the currents), but they do not tell how to control them: the designer can freely choose the controller type. However, these approaches with models give the references of the currents to be controlled, and the model used brings a quality to these references, according to the precision of the model used. Moreover, these are the references that must be synchronized with the help of the position sensor (“self-control”).

According to the machines‘ characteristics and the precision of the models used to represent them, various alternatives will intervene:

– The first alternative relates to the difference between non-salient pole machines and salient pole machines. The difference comes from the geometries used to build these machines. Thus, in the case of the synchronous motors with “magnets installed on the surface”, the air gap is constant and the machine is with non-salient poles. It is then easy to use a three-phase model in the three-phase a-b-c reference frame. On the other hand, for “buried magnet” motors (in the rotor), the machine is with salient poles and the three-phase model is appreciably more complex, whereas the Park model (d−q) leads to a remarkably simple model. It is necessary to be aware that the simpler a model is, the easier it is to design efficient controls. Of course, we can give a model in d-q of a non-salient pole motor (the model is then even simpler), but we can see here that a constructive structure can have effects on the choice of modeling. The modeling itself has effects on the control design.

– The second important variant is also constructive: the alternating current machines are known as “well built” if they obey the Park hypotheses (i.e. to the hypotheses implicitly used for the conventional modeling in the d-q reference frame). It is necessary to distinguish the machines with “sinusoidal field distribution” and the ones with “non-sinusoidal distribution”. The most well known of these last machines is the “trapezoidal” back-EMF machine. These machines have excellent physical performances in terms of “mass torque”. They are thus very popular among designers and users. But a precise model is difficult to write and exploit for the control. It is then common to accept approximations:

- on the model level (accepting a model limited to the first harmonic, in order to find the model of the sinusoidal field distribution machines);

-  or on the control level (supplying the stator by square wave currents, for example): we will show (Chapters 2 and 3) supply and control examples of sinusoidal field distribution machines with extensions to the non-sinusoidal cases. There too, modeling has effects on the machine control strategy.

– Another variant is the existence of saturation: the Park model assumes that we operate in linear (unsaturated) regime, which is legitimate for the machines with magnets installed on the surface, because they have a large air gap and they only saturate a little. But this hypothesis can be faulty with small air gaps machines. Obviously, a saturated model of the synchronous machine is extremely complex – and not easy to use for the controls‘ design (let us recall that the conventional models of synchronous saturated machines are limited to the steady state, whereas the control laws are conceived with dynamic models). The conventional control laws are generally unaware of the saturation effect. However, there are extensions to the methods, in order to discuss the case of saturated machines, which will refer to them ([STU 01]).

- The last alternative concerns the third hypothesis attached to the Park modeling: the latter assumes that the machine is “symmetrical”, “balanced”, or “respects the circularity hypothesis”. This means that the three stator phases are identical and simply shifted within the space of an electrical angle of 2 · π/3. We will consider that this hypothesis is verified (the construction cannot however be perfect), except in the case of a machine “at fault”, victim of a fault. This case will be discussed in a specific chapter of another book of this EGEM (Electronique-Génie Electrique-Microsystèmes: Electronics-Electrical Engineering-Microsystems) treatise [FLI 10].

The other large control family relates to the absence of a mechanical sensor. We have already stated that the controls without sensors will be covered in Chapters 8 and 9 of this book.

1.2.3. Control properties

This is the opportunity to specify one of the main advantages of the synchronous motor, when it is controlled with mechanical sensors (speed and position). This is a fundamental hypothesis of this chapter. With these sensors, all the state variables are measurable (or calculable). The controls can thus profit from a “complete state feedback”: the designer can place all the system poles in closed-loop exactly where he wants. In practice, we will use this property. In addition since the criteria selection to design the current and speed controllers is extremely broad, we have standardized our own account by limiting ourselves to only one method. this method systematically seeks to impose “critical damping”, by forcing the closed-loop system to have its n time-constants all equal: 1/(1 + τ_bf · p)ⁿ. This is in fact a “robust poles placement”, the n poles being all equal and real: p_i = −1/τ_bf (1 ≤ i ≤ n). Thus, the adjustment is limited to the choice of only one parameter, the time-constant τ_bf . To choose its value, we define a time-constant supplying a legitimate order of magnitude, that we call the “reference time- constant”, noted here τ_Ref. We choose the “electric time-constant” for the current loops (Chapters 2 and 3) and the “electromechanical time-constant” for the speed loops (Chapter 4). Then we set out: τ_bf = τ_Ref /λ . It is then sufficient to choose only one parameter - the λ coefficient (without dimensions) - to determine all the controller parameters.

In this account, we had to make choices, because the synchronous motor control putting in motion an “axis” can have alternatives and particular points. Thus, in the controls that will be presented, first in torque (in practice, in current), then in speed, we only consider “proportional” controls (known as “P”), to which we add an integral effect. There are then two alternatives. The most widespread is the “proportional-integral” variant (known as “PI”), but we prefer the less common, but now conventional alternative, known as “IP” (thus “integral-proportional”). However, for the current controls as well as for the speed controls, we had to show examples of “advanced” controllers: the “resonant controller” for the current controls in the a-b-c reference frame (Chapter 2), and the controller with “observer” for the speed controls (Chapter 4).

Let us talk about the advantages of the “IP” controller. It does not introduce a zero in the transfer function in closed-loop (thus its other name, “PI without zero”). Therefore, when we test its performances on a step test, it does not introduce an additional overrun compared to those that the poles can naturally introduce. The choice of the λ parameter (imposing the poles) thus completely defines the performances in terms of overrun and response time. Moreover, the IP controller can be designed and implemented in two stages: the proportional loop first and then the integral loop – which is very practical during implementation. We will see that there is strong logic behind this reasoning. Indeed, it is natural and simple to conceive a proportional loop (either alone, or completed by “compensations”). It is easy to then add the integral loop to increase the robustness of the control (insensitivity to errors or to lack of knowledge of the models).

1.3. Descriptions and physical modeling of the synchronous motor

1.3.1. Description of the motor in preparation for its modeling

We will not detail the physical description and the mathematical modeling of the synchronous motors, studied in detailed chapters ([MAT 04], [SAR 04]) in other books of this EGEM treatise ([LOU 04a], [LOU 04b]) and specialized books ([GRE 01], [CHA 83]). Let us only point out the essential elements of these machines‘ structures:

– the stator is usually three-phase and is built in order to have “non-salient poles”;

– the rotor has the most variants. We will consider two main variants:

-  the excitation can be carried out by single-phase winding (Figures 1.1 and 1.3) or by permanent magnets (Figure 1.2, a very frequent case in low and average power),

- the rotor can be built in order to have “non-salient poles” (constant air gap, Figures 1.1 and 1.2) or “salient poles” (variable air gap, Figure 1.3).

Figures 1.1 to 1.3 give representations of three typical examples of synchronous machines, used as motors.

The case of the machine with salient poles and wound excitation (Figure 1.3) can be considered as the “generic” case, likely to represent the various alternatives. We can easily bring it back to the other cases by making conventional simplifying hypotheses:

– we find the case of magnet excitation by making the hypothesis of the “Amperian currents”. Seen from the outside, the magnet is equivalent to an air coil crossed through by a constant current;

– we will see that the saliency introduces into the models a specific parameter (noted L_s2 ) and that it is sufficient to pose that this parameter is equal to zero, to model a non-salient pole machine.

It is then interesting to consider a “symbolic representation” (Figure 1.4.), far away from the physical representation, but which makes it possible to easily write the mathematical machine model.

1.3.2. Hypotheses on the motor

The conventional motor modeling is based on the conventional hypotheses of the “well-built machine in the meaning of Park”. They will be formalized by the equations in section 1.3.5. We summarize them here:

– “first harmonic” hypothesis: in the air gap, the magnetic field has a “sinusoidal field distribution according to the space variable”. This will clearly appear in the form of inductance matrix expressions [1.5] and [1.7];

– linearity hypothesis: flux is proportional to the currents that created them, as is clearly shown in equations [1.1] and [1.2];

– “symmetry” or “circularity” hypothesis: the three-phase windings are identical and simply shifted within the space of an electrical angle of 2 · π/3. This will appear

in equations such as [1.4] and [1.5];

– in fact, they are hypotheses ad hoc, helping us to write the models by combining simplicity with a good efficiency (they are conventional hypotheses legitimated by the experience of the scientific and industrial community). We thus neglect: the skin effect, eddy currents, etc. There are more advanced models, but they will not be part of this account.

Most of this account will be devoted to machines respecting these hypotheses. We will however give general results applicable to the “non-sinusoidal” field distribution machines with non-salient poles (Chapter 2) and with salient poles (Chapter 3). Some chapters of books to come in this EGEM treatise will discuss variable reluctance machines and synchronous reluctant machines [TOU 11].

1.3.3. Notations

Under these conditions, the electric or magnetic “natural” variables (current, voltages, flux) are three-phase at the stator and single-phased at the rotor (excitation). We will use vectorial notations for the stator three-phase magnitudes, with the index “3”. We will use the index “f ” for the excitation (for “field”). We detail them in Table 1.1.

1.3.4. Main transformation matrices

To have compact writings giving way to easy calculations, later on we will need matrix notations (see [SEM 04]) using the Clarke and Concordia sub-matrices, as well as the rotation matrix defined by Table 1.2, where the I₂ matrix is the matrix dimension unit 2x2.

1.3.5. Physical model of the synchronous motor

With these hypotheses and notations, the machine is physically completely modeled with the equations presented in this section. We start with the flux equations:

[1.1]

[1.2]

These equations are the most fundamental equations. We assume the constant parameters (clean and mutual inductances) to be known, such as M₀ , L_f , and those intervening in the inductances matrices. Those have as expressions:

[1.3]

where (L_ss0) represents the constant part and (L_ss2 (θ)) the variable part. The latter is due to the presence of a term L_s2 describing the saliency effect (air gap variation):

[1.4]

[1.5]

Note.– for a non-salient pole machine, as seen previously, the saliency term disappears. It is then sufficient to set out:

[1.6]

Finally, (M_sf (θ)) has the following expression, also given in a second “factorized” form to give a compact writing example that we will use a lot later on:

[1.7]

Then, we write the equations at the voltages:

[1.8]

[1.9]

where the R_s resistances of the stator phases windings are assumed to be, by assumption, all identical; R_f is the excitation winding resistance. Resistances are all assumed to be constant.

The last equations to be considered are relative to the torque. [MAT 04] and [LOU 04c] have shown that the determination of the electromagnetic torque goes by the preliminary determination of the magnetic co-energy. The latter is expressed:

hence:

[1.10]

We deduce from it a general expression of the electromagnetic torque (we will observe that the result is a scalar):

[1.11]

We see that expression [1.11] contains two very distinct terms:

– the first term describes the saliency effect. It can be the only one in the case of the “synchronous reluctant motors”, which do not have an excitation, but a strong saliency. It is equal to zero in the case of “non-salient pole machines”. It contributes to the torque, often in a secondary way, in the “synchronous salient pole motors”;

– the second exists for all the excited motors (by winding or magnets). It is then often the dominant term. This is why the excitation winding is classically called: the “field system”;

– this model ignores the cogging torque: it is a frequently accepted (and largely justified) hypothesis. This torque C_d (0) remains when the stator is not supplied (for

example, it can be due to the interaction between the stator teeth and the rotor magnets). However, in Chapters 2 and 3, we will present general summary methods, which can take it into account. In this case, the torque has as a general expression:

[1.12]

1.3.6. The two levels voltage inverter

We assume that the motor is supplied by a two level voltage inverter (Figure 1.5) classically controlled in pulse width modulation (PWM). We do not detail the description and modeling, which are the subject of specific books of this treatise ([MON 11]) and of specialized books [LAB 95].

Some chapters of [LAB 04] and [LAB 11] have shown that we can model the inverter as a three-phase voltage amplifier. We note (u₃) = (u_a u_b u_c)^t , the (u₃) = (u_a u_b u_c)^t, the

control signal. We can define a constant gain G₀ so that:

[1.13]

The supply by inverter imposes that the total current is equal to zero:

[1.14]

In the modeling framework with the three conventional hypotheses (linearity, first harmonic, circularity), this authorizes us to set out that the zero-sequence components of all the variables are equal to zero. If we want to exploit the zero- sequence component current properties (in particular in Chapter 2, section 2.5), we will have to use an adequate inverter (for example an inverter sometimes called “3 H” and which is three times single-phase).

1.3.7. Model of the mechanical load

We will not develop here all the specific problems of the “axis control” assuming the recognition of mechanical phenomena, which can be very complex ([HUS 03]). We will limit ourselves to the essential properties presented in Chapter 4. We choose a conventional equation to describe the dynamics of the axis mechanical part:

[1.15]

to which it is necessary to add the position dynamics, since it intervenes in the Park transformation [1.66] (consequently, unlike the direct current motor, the speed regulation of a synchronous motor involves the position):

[1.16]

The parameters have the conventional meanings: J represents the rotating parts inertia, f the coefficient relating to viscous frictions, C_ch the load torque assumed to be piece-wise constant.

1.4. Modeling in dynamic regime of the synchronous motor in the natural three-phase a-b-c reference frame

1.4.1. Model of the machines with non-salient poles and constant excitation

1.4.1.1. General properties

Historically, designers often worked on non-salient pole machines and used as a priority models directly resulting from the equations written in the natural a-b-c reference frame, because we could, at least at first, bring it back to three traditional single-phase equations (“three-single-phase” system), while profiting from the fact that the total current is equal to zero when the motor is supplied by the two-level voltage inverter (section 1.3.6). In addition, we will consider the classical case of the magnet excitation machine, deduced from model [1.1], while assuming that the excitation current is constant. To introduce a three times single-phased model, we introduce the magnitude ψ_af (first component of (ψ_3f) = (ψ_af ψ_bf ψ_cf)^t ). Then, the electric equation relative to the first phase is written:

[1.17]

where we revealed the cyclic inductance L_a (whose expression depends on the field distribution), the phase resistance:

[1.18]

and the counter electromotive force (due to the only excitation effect, hence the index f) e_af :

[1.19]

We can extend this definition to the three phases. The expressions of the three- phase back-EMF (e_{3 f} ) = (e_af e_bf e_cf )^t can be written under a vector form:

[1.20]

In the following, we will often take a look at the flux derivatives, created by the excitation in stator windings. They are defined by:

[1.21]

In practice, the flux derivative corresponds to the induced back-EMF, divided by the rotation speed; they are very useful magnitudes. An assessment of the powers defines the power converted into mechanical power (noted P_m ) and gives the relation between the electromagnetic torque (noted C_em ), the currents and the back-EMF. Indeed:

[1.22]

[1.23]

or, from [1.21] and [1.23]:

[1.24]

These general properties can be specified in the case of a sinusoidal distribution machine.

1.4.1.2. Case of a synchronous machine with sinusoidal field distribution

All formulas from [1.17] to [1.24] remain usable, but it is more practical to clarify a certain number of results. We can specify the expression of cyclic inductance:

[1.25]

and introduce a Φ_f0 coefficient, so that:

[1.26]

where:

[1.27]

Then the counter electromotive force e_af has as an expression:

[1.28]

For the following, we will set out:

[1.29]

with:

[1.30]

The expressions of the three-phase back-EMF can be written under a factorized form:

[1.31]

and the flux derivatives created by the excitation in stator windings have as expressions:

[1.32]

In the sinusoidal case considered here, Figure 1.6 gives the respective flux speeds and flux derivatives in reduced magnitudes. Let us note that, on these curves, the back-EMF induced in reduced magnitudes are identical to flux derivatives:

[1.33]

For the synchronous sinusoidal field distribution machine, we can give a first detailed expression of the electromagnetic torque deduced from [1.24]:

[1.34]

1.4.2. Exploitation of the model in the a-b-c reference frame in sinusoidal steady state, electromagnetic torque

1.4.2.1. Expression of the electromagnetic torque

The sinusoidal continuous rating is a very important classical case, in which we can clearly see the fundamental properties. Its properties are conventional [LED 09]. The average torque is not equal to zero, if the machine is supplied with three-phase sinusoidal balanced direct currents. These properties can be summarized as follows:

– the electric variables‘ angular frequency is imposed by the supply and its value is denoted ω ;

– the rotation speed is then:

[1.35]

– the position thus verifies:

[1.36]

– to write the currents, it is interesting to define a phase displacement α ² as follows:

[1.37]

Then, the torque equation can be written, with the use of a torque coefficient, noted K_abc , in formula [1.38]:

[1.38]

[1.39]

1.4.2.2. Electromagnetic torque optimization: self-control

In the upcoming controls of the synchronous machine that we will see in Chapter 2, the phase displacement α is imposed and is then called the “delay angle”. It is common that we seek to maximize sin (α), in order to minimize the efficient amplitude I₁ (consequently, we minimize the Joule losses for a given torque). This optimal control thus imposes:

[1.40]

We have just defined, on this simple example, “the self-control”: the currents are synchronized on the position θ (see [1.36]). The “self-control” concept is thus immediately deduced from the synchronous machine properties. We justify it here from its modeling. The optimization that we have just presented is a control (“inverse model”) immediately deduced from equation [1.38] (“direct model”).

Figure 1.7 shows that the electromagnetic torque, given by [1.24], verifies [1.38] in the case [1.40]: the current is in phase with the flux derivative (in reduced magnitudes, the curves are superimposed). The torque is quite constant, without oscillations, and for a given amplitude I, it is at its maximum.

1.4.3. Extensions to the case of non-sinusoidal field distribution machines

1.4.3.1. Case of trapezoidal field distribution machines

The sinusoidal distribution machines are known as “well built in the Park sense”. These machines are intended to be naturally fed by sinusoidal currents. However, for legitimate technological reasons (maximization of the torque mass criterion for example, see [MUL 06]), field distribution can be non-sinusoidal. The most frequently described case is the “trapezoidal” distribution (see bottom curves of Figure 1.8: series of “plateaus” and slopes). The most useful magnitude is the flux derivative (equal to the back-EMF divided by the speed Ω, i.e. ψ‘_af = e_af/Ω). On a quarter of a period, its expression is:

[1.41]

where:

[1.42]

and E_m is the value of the back-EMF during the plateau.

The developments in Fourier series of the flux and its derivative relative to the first phase are then:

[1.43]

[1.44]

The back-EMF can also be written:

[1.45]

In formulas [1.43] to [1.45], we always factor in (thus on the left of the sign “ ∑ ”) the amplitude expression of the “first harmonic”, i.e. the fundamental (term classically noted a₁). The amplitude of the term on the right of the sign “ Σ ” is thus the ratio of the harmonic order 2·k+1 (term classically noted a_2k+1) divided by this amplitude, i.e. a_2k+1/a₁ (see also the table of the numerical values in appendix section 1.7.1). The example in Figure 1.8 corresponds to a conventional case, the one where δ = π/6. We will see later on (Chapter 2, section 2.2.2.) that this machine can be simply fed by a system of three-phase square wave currents. We observe that the flux of the trapezoidal machine consists of line segments (when the plateau of the derivative, and thus of the back-EMF, is constant), constituting saw teeth. These segments are linked by parabolic arcs (when the derivative varies linearly).

1.4.3.2. Case of non-sinusoidal field distribution machines

The effective cases are often those illustrated in Figure 1.9.

In this account, the non-sinusoidal examples have been chosen so that they all have the same first harmonic (or fundamental). The latter is identical to the sinusoidal model illustrated in Figure 1.7. It can be regarded as the model close to the first harmonic of the non-sinusoidal machines of Figures 1.8 and 1.9. It is frequent that these non-sinusoidal machines are studied within the meaning of the first harmonic, but, we can perform much more precise studies relative to all the harmonics, as we will show in Chapter 2.

In the example presented in Figure 1.9, flux and flux derivatives are modeled by Fourier expansions [1.46] and [1.47] (Note: the index “ns” indicates the non- sinusoidal case).

[1.46]

[1.47]

Figure 1.10 compares the flux derivatives for the three cases: sinusoidal, non- sinusoidal, trapezium (δ = π/6). If we consider that the “non-sinusoidal case” is the general case, we can admit that the trapezoidal and the sinusoidal case are two levels of approximation and idealization.

1.5. Vector transformations and dynamic models in the α-β and d-q reference frames (sinusoidal field distribution machines with non-salient and salient poles)

1.5.1. Factorized matrix modeling

We consider the case of sinusoidal field distribution machines. The model described by equations [1.1] to [1.11] is strongly non-linear. We will see that the controls designed in this (natural a-b-c) reference frame are, either not very efficient, or difficult to implement.

Moreover, it is conventional to carry out “transformations”, i.e. changes of reference frames simplifying the equations‘ form, making them more suited to the design of performance controls.

These changes of reference frames have been detailed in [SEM 04] and [LOU 04d].

They are also the subject of specialized books ([LES 81], [CHA 83]). The zero- sequence components are generally assumed to be equal to zero in our studies, when the supply is obligatorily made by a three-phase voltage inverter on two levels.

When the zero-sequence components are not equal to zero, we point it out explicitly (for example, in Chapter 2, section 2.5, in particular section 2.5.3).

Under these conditions, we can successively define two transformation types: the Concordia transformation and the Park transformation. We will not detail the calculations, but they are very easy to carry out when we have the following “factorized” forms of the inductance matrices:

[1.48]

[1.49]

where the constant parameters are:

[1.50]

[1.51]

[1.52]

[1.53]

Note.– For a non-salient pole machine: L_d = L_q = L_cs , because L_s2 =0 .

[1.54]

1.5.2. Concordia transformation: α-β reference frame

With the normalized Concordia transformation, we go from the natural a-b-c reference frame to an adequate two-phase reference frame (called “ α - β”).

It does not preserve the signal amplitudes, but it preserves the power. It is thus defined for all the (electric or magnetic) variables:

[1.55]

where:

[1.56]

After application of this transformation, we can define a diagram of the two- phase machine (known as “α−β”), equivalent within the Concordia meaning (see Figure 1.11).

The Concordia transformation gives the following results:

– for the equations at the flux:

[1.57]

or:

[1.58]

[1.59]

[1.60]

– for the equations at the voltages:

[1.61]

[1.62]

[1.63]

Let us note that the equations relative to the zero-sequence components are completely uncoupled from the equations relative to the two-phase variables. For the following, it will be interesting to define notations as:

where (ψ_2f (θ)) is relative to the flux created by excitation. We will also note the two-phase back-EMF:

– for the magnetic co-energy and the electromagnetic torque:

[1.64]

[1.65]

1.5.3. Park transformation, application to the synchronous salient pole motor

Park transformation ([PAR 28a, PAR 28b, PAR 29, PAR 33]) is a rotation of the two-phase reference frame, making it possible to align it with the rotor axis, the new axes then being called “d-q”:

[1.66]

or:

[1.67]

The zero-sequence components are identical in the α-β and d-q reference frames. After transformation, we can define a diagram of the two-phase machine, equivalent within the Park sense (see Figure 1.12).

The equations of the synchronous machine after the Park transformation are given by the following relations:

– first for the flux:

[1.68]

– then for the voltages:

[1.69]

or:

[1.70]

where:

[1.71]

To represent the magnet machines, we assume that the excitation current is constant and we set out:

[1.72]

Lastly, for the electromagnetic torque:

[1.73]

For a non-salient pole machine, this equation can be written:

[1.74]

with:

[1.75]

Equations [1.13], [1.55], [1.66], [1.15] to [1.16] and [1.68] to [1.73] constitute the dynamic mathematical model of the axis driven by the synchronous motor. From it, we can give an “input-output diagram”. The latter is represented by Figure 1.13. This complete dynamic model will be our “simulation model” which, completed by the equations of the controllers, will help us to judge the controls performances.

It is necessary to distinguish this “dynamic model” (which is as complete as possible) from the simplified “control models”, that we will use in Chapters 2 to 4 to conceive (or “to design”) often simplified controllers.

1.5.4. Note on the torque coefficients

We can also note:

[1.76]

to insist on the analogy between “the q axis” of the synchronous machine and “the armature” of the direct current motor. Let us recall that the torque coefficient defined in the framework of the a-b-c modeling (see formulas [1.38] and [1.39]) is noted K_abc and that it verifies the property:

[1.77]

1.6. Can we extend the Park transformation to synchronous motors with non- sinusoidal field distributions?

The Park transformation has excellent properties. It leads to the simplest possible writing of the equations of the machine in dynamic regime and to the determination of efficient control laws (this will be the object of Chapter 3). But, can we extend it to machines not answering the three conventional hypotheses (linearity, circularity, sinusoidal field distribution)? Strictly speaking, no (see [XIA 89]). But if we admit to giving up some properties, we can define transformations (described as “extended Park”) solving some very important problems in practice, such as the direct determination of the currents imposing a desired torque, without ripple and by minimizing the Joule losses. In this spirit, optimizations will be the subject of the last sections of Chapters 2 and 3 of this book. In this section, we will limit ourselves to the synchronous non-salient pole motors, with constant excitation, without saturation nor cogging torque, but with a non-sinusoidal field distribution, such as the machine presented in section 1.4.3. The fundamental electric equation is equation [1.8], but the flux expression (ψ)₃) is no longer given by the first harmonic theory, but is given by a more general expression:

[1.78]

For example, the flux created in the stator by excitation (ψ_3f (θ)) is given by the curves of Figures 1.8 and 1.9. The torque is always defined by the general formula but as the machine is with non-salient poles, we can already limit it to the expression deduced from the power evaluation already seen, (see [1.22] and [1.23]) where the back-EMF due to the excitation are given by equations [1.19] to [1.21]. Several strategies can contribute to defining extensions of the Park transformation. We will expose it as an “inverse problem”: how do we determine the stator three-phase currents ( i₂ ) = ( i_a i_b i_c )^t, so that we obtain a desired torque C_{em wis} ? The equation to be solved can be deduced from [1.20] to [1.23]:

[1.79]

where we reveal the flux derivatives with respect to the position: (ψ’_3f) = (ψ’_af ψ’_bf ψ‘_cf)^t = (e_3f)/Ω. Solutions to this problem have been proposed by digital optimization ([MAR 92]).

We choose here an analytical approach, that we consider as a heuristic extension of the Park transformation ([GRE 93], [GRE 95], [GRE 97], [GRE 98], a method reused in [MON 04]). For more details, we can read the chapter in [GRE 04]. We give here a short presentation.

The conventional Park transformation (see section 1.5.3) has several “good” properties: the reference frames first deduced by the Clarke or Concordia, transformation and then by an angle rotation p₁ · θ are orthogonal. Moreover, we decide to choose a normalized transformation, preserving the power. In addition, if we examine [1.70], we see the meaning of the results [1.71]: the back-EMF term due to the excitation is equal to zero on the d axis.

Lastly, for a non-salient pole machine, the torque expression is reduced to C_em = p₁ · (e_qf/Ω)·i_q (see [1.74] and [1.75]). Only the q axis current intervenes: it is a very important property, because it is a very powerful tool to determine the feed currents (inversion of the model).

By hypothesis, we thus choose to preserve:

– a property similar to [1.71]: the term e_df is equal to zero;

– the torque is written with the only current i (as in [1.74]s). There will only be one unknown factor (this current), solution of the single equation [1.79];

– and here, we assume that the machine is supplied “with three wires”. We thus do not consider the zero-sequence component currents effect.

Under these conditions, the extension of the Park transformation is described by the following reasoning.

FIRST STAGE.– Concordia transformation

All the useful three-phase magnitudes are transformed by [1.56] into two-phase magnitudes “ α - β ”. We illustrate this transformation in Figure 1.14, presenting the Concordia components of the excitation flux derivatives, which verifies the following relations with the back-EMF:

[1.80]

SECOND STAGE.–Conventional Park transformation

If we apply the angle p₁ · θ rotation [1.66], the d-q components of the flux derivatives (proportional to the back-EMF) are given in Figure 1.15.

We observe that this machine, which does not have sinusoidal field distribution, does not verify the properties [1.71]: (a) the back-EMF of the d axis: e_df=Ω·dψ_df/dθ is not equal to zero (this back-EMF fluctuates around 0), and (b) e_qf = Ω · dψ_qf /dθ is not constant. This is the first property (a) which for us is annoying, since the torque then depends on i_d and we do not profit from the very useful property [1.71]. Thus, the effective following extension.

THIRD STAGE.–Extended Park transformation

We propose to replace the conventional rotation p₁ · θ with an “extended rotation” of angle p₁ ·θ_e verifying:

[1.81]

where the angle μ(p₁ · θ) is chosen so that, in this new reference frame, the axis component “extended d” of the back-EMF, noted e_de, is equal to zero. We thus have:

[1.82]

We must solve:

[1.83]

We cancel the e_de component, if we choose the angle μ(p₁ · θ) so that:

[1.84]

And then, the e_qe component is given by:

[1.85]

Lastly, the torque is given by:

[1.86]

Extended Park components of the currents verify:

[1.87]

This reasoning gives us several results. First, we can illustrate it by the back- EMF form (Figure 1.16) in the extended reference frame defined by [1.82] or [1.83]. In Figure 1.16 we observe the following properties: the angle μ(P₁·θ) is a variable, whose oscillations show the effect of the harmonics contained in the back-EMF of this non-sinusoidal field distribution machine.

Visually, it appears that the essential element is found in 6^th order harmonics, as we will show in Chapter 2.

We observe that, as we chose it, the component of the extended d axis, e_de (p₁ ^· θ) , of the back-EMF is thus equal to zero. The component of the extended q axis, e_qe (p₁ ·θ) of the back-EMF is not constant, as in the case of the conventional Park transformation. We observe however that this back-EMF fluctuates with not too high amplitudes around its mean value. These forms are much simpler than those observed after the conventional Concordia or Park transformations.

Then, we can write a dynamic model in the extended Park reference frame, defined for all the variables by:

[1.88]

The back-EMF in the extended Park reference frame verifies:

[1.89]

and we can show:

[1.90]

This model is more complex than the conventional Park model, but simpler to use than the models with alternative wave forms (three-phase or two-phase). This model can be exploited for the control design of currents in the extended reference frame, similar to those presented in Chapter 3.

Finally, we can easily determine (it was one of the sought-after goals) an optimal current imposing the desired torque: to minimize the Joule losses, we choose a current reference of the extended d axis equal to zero (it does not intervene in the torque expression) and to define the current reference of the extended q axis, we invert [1.86]:

Various ways of exploiting this result are possible (as just seen, to design a torque control for example). We will limit ourselves here to visualizing the optimal currents (Figure 1.17) in the extended reference frame and in the natural three-phase reference frame, because indeed, we can determine it by:

[1.91]

Figure 1.17 gives the expected results: the extended q axis current ( i_{qe _ ref} ) fluctuates with a low amplitude around its mean value (it compensates for the fluctuations of the back-EMF of the same axis). The three-phase currents are alternating currents whose distortion (compared to the conventional sinusoids) compensate for the non-sinusoidal effects of the back-EMF.

The bottom figure is a simple check: we observe the mechanical power converted at the same time in the extended reference frame and in the natural three- phase reference frame:

[1.92]

We check that these powers are quite constant (and identical). We observe that some properties still cause problems. Thus, it is necessary to control a current i_qe containing ripples. There are other alternatives, resulting from other approaches ([AKA 93]), leading to the definition of “denormalized transformations” ([YAL 94], [GRE 98], [GRE 04]), where the component on the q axis does not have any ripples.

In addition, the approach presented here is restricted by various hypotheses (non- salient pole machine, no saturation, no cogging torque, no zero-sequence component current, etc.). We will see in Chapters 2 and 3 more general methods for removing most of these restrictions.

1.7. Conclusion

We presented most of the modeling of a synchronous motor; mainly the usually considered motor, which answers to the conventional hypotheses: symmetry, linearity, first harmonic. However, we presented some extensions of the non- sinusoidal field distribution machines (in particular trapezoidal), and we will present some others, more powerful ones, in Chapters 2 and 3.

Torque controls can be based on various models: in particular on the three-phase or two-phase model “α-(β” (Chapter 2) or on the Park model (Chapter 3). These models lead to the determination of the references of the currents necessary to the determination of the optimal torque (without ripple, with minimal Joule losses) and the structure of the current controls. To take into account the various approaches of the synchronous motor controls, it is thus necessary to know these various models.

1.8. Appendices

1.8.1. Numerical values of the parameters

The examples in Chapters 1, 2, 3 and 4 concern a motor whose parameters are as follows.

Poles pair	P1 =3
Average self inductance of a phase	Ls0 = 33 mH
Average cyclic inductance	Lcs =49.5 mH
Second order harmonic	Non-salient poles: Ls2 = 0 Salient poles: Ls2 =5 mH
Direct axis inductance	Non-salient poles: Ld = Lcs = 49.5 mH Salient poles: Ld = 57 mH
Quadrature axis inductance	Non-salient: poles: Lq = Lcs = 49.5 mH Salient poles: Lq = 42 mH
Excitation flux by phase: 1st harmonic and superior harmonic (in Wb)	Φf1 = Φf0 = 0.255 Wb Φf3 = 0.018, Φf5 = 0.00112, Φf7 = -0.00146, Φf9 = -0.00125
Voltage coefficient (relative to a root mean square voltage made between phases)	Kabc_U_eff =98V/1000 tr_mn
Resistance by phase	Rs = 12.25 Ω
Nominal torque	CN = 2.3 Nm
Rated RMS current	IN =1.42 A
Rated speed	3000 tr/min
Inertia	J = 0.01 kg.m2
Viscous friction	Inertia case: f = 0 N.m/rad.s-1 Trial robustness case: f = 0.01
Current sensor	ki = 0.1 Ω
Speed sensor	ki =0.06 v/rad·s−1
Converter gain	G0 =12

Table 1.3. Numerical values of the parameters of the synchronous motors taken as an example

1.8.2. Nomenclature and notations

1.8.2.1. General notations

– p₁ : number of pairs of poles;

– t: time;

– s: index of the sinusoidal case;

– tr: index of the trapezoidal case;

– ns: index of the non-sinusoidal case.

Three-phase variables in the natural reference frame:

−(i₃ ) = (i_a i_b i_c )^t: stator currents;

−( v₃ ) = (v_a v_b v_c )^t: stator supply voltages;

−(ψ|₃) = (ψ_a ψ_b ψ_c )^t: stator flux;

−(ψ_3f) = (ψ_af ψ_bf ψ_cf )^t : flux created by excitation in the stator phases;

−(ψ’_3f) = (ψ’_af ψ’_bf ψ’_cf )^t : derivative (with respect to the position) of the flux created by excitation in the stator phases;

−(e_3f) = (e_a e_b e_c)^t: stator counter electromotive forces (back-EMF).

1.8.2.2. Single-phase variables in the natural reference frame (usually: first phase) and parameters

1.8.2.2.1. Stator variables

– i_a: current;

– I, I_ref: efficient current amplitude (case of the sinusoidal steady state), its reference value;

– v_a: supply voltage;

– ψ_a: flux embraced by the phase;

– e_af : counter electromotive force (back-EMF);

– E_m : amplitude of the voltage “plateau” (case of the machines with trapezoidal back-EMF);

– δ: half angle during which the back-EMF fluctuation is linear (case of the trapezoidal machines with back-EMF);

–  I_m : amplitude of the current square wave (case of the machines with trapezoidal back-EMF);

– k: variable to define the orders of the harmonics 2·k + 1 of the non- sinusoidal back-EMF machines;

– φ_f1, φ_f3, φ_f5, φ_f7, φ_f9: harmonics coefficients of the Fourier expansion of the flux in a stator phase, in the non-sinusoidal case;

– φ_f0 = |φ_f1|: 1^st harmonic amplitude of the flux created by the excitation in a stator phase;

– Ψ_af : flux created by excitation in the first stator phase;

– ψ‘_af : derivative (with respect to the position) of the fluxi|ψ_af created by excitation;

– ψ_{af_s}, ψ‘_{af_s} : flux and derivative (with respect to the position) of the flux created by excitation, while specifying: in the sinusoidal case;

– ψ_{af_tr}, ψ‘_{af_tr} : flux and derivative (with respect to the position) of the flux created by the excitation, while specifying: in the trapezoidal case;

– ψ_{af_ns} , ψ’_{af_ns} : flux and derivative (with respect to the position) of the flux created by excitation, while specifying: in the non-sinusoidal case;

– K_Ω: back-EMF coefficient;

– K_abc: torque coefficient, case of the sinusoidal steady state;

−K_tr: torque coefficient, case of the machine with trapezoidal back-EMF supplied with square wave currents;

– α , α_opt : control angle, its optimal value.

1.8.2.2.2. Stator parameters, inductances and resistances

– L_SS(θ) or L_ss(p₁ · θ): stator inductances matrix;

– (L_ss0) constant matrix terms of the self and mutual stator inductances;

– M_sf(θ) or M_sf(p₁ · θ): matrix of the stator-rotor mutuals (excitation);

– L_f : self inductance of the excitation winding;

– L_s0, M_s0: self inductances, mutual of the stator phases;

– L_s2: amplitude of the second harmonic of the stator inductances;

– M_f0: amplitude of the stator-rotor mutuals (excitation);

– L_α = L_cs : cyclic inductance of a stator phase (non-salient poles case);

– R_s : resistance of a stator phase;

– τ_es = L_cs/R_s : electric time-constant of a stator phase.

1.8.2.2.3. Parameters and variables associated with excitation

– R_f : resistance of the excitation winding;

– i_f : current;

– v_f : voltage;

– ψ_f : flux.

1.8.2.3. Variables, vectors and matrices, after the Concordia and Park transformations

– C₃₂: Clarke sub-matrix;

– T₃₂: Concordia sub-matrix;

– P(ξ): angle rotation of ξ

– I₂: 2^nd order unit matrix;

– J₂ : angle rotation matrix −π/2;

three-phase stator variables (currents, voltages, flux, flux derivative, back-EMF);

two-phase variables after the Concordia transformation;

, two-phase vectors (voltages, currents, flux) after the Concordia transformation;

amplitude of the stator-rotor mutual (excitation) after the Concordia transformation;

two-phase variables after the Park transformation;

two-phase vectors (voltages, currents, reference current and flux) after the Park transformation;

two-phase variables after the “extended Park transformation”;

– (x_f) the index “f indicates that it is the excitation effect on stator windings (induced flux and back-EMF);

– L_d,L_q: stator inductance of the d axis and the q axis;

amplitude of the excitation flux created in a stator phase after the Park transformation;

– K = K_q: torque coefficient after the Park transformation (concerning the q axis).

1.8.3. Acknowledgments

We warmly thank G. Feld for his help.

1.9. Bibliography

[AKA 93] AKAGI H., NABAE A., “The P-Q theory in the three-phase systems under non- sinusoidal conditions”, ETEP, vol. 3, p. 314-326, January 1993.

[BEN 07] BEN AHMED H., BERNARD N., FELD G. , MULTON B., “Machines synchrones - Principes généraux et structures, Techniques de I’Ingénieur, Paris, D 3520 - Machines synchrones - modélisation en régime permanent”, Techniques de I’Ingénieur, Paris, D 3521 - Machines synchrones - fonctionnement en régime permanent, Techniques de I’Ingénieur, Paris, D 3522, May 2007.

[BOS 86] BOSE B. K., Power Electronics and AC Drives, Prentice-Hall, Englewood Cliffs, United States, 1986.

[CAN 00] CANUDAS DE WIT C. (ed.), Commande de moteurs asynchrones 1: Modélisation, contrôle vectoriel et DTC 2: Optimisation, discrétisation et observateurs, Hermès, Paris, 2000.

[CAR 95] CARON J.-P., HAUTIER J.-P., Modélisation et Commande de la machine asynchrone, Technip, Paris, 1995.

[CHA 83] CHATELAIN J., Machines électriques, traité d‘Electricité vol. 10, EPFL, Presses Polytechniques Romandes, Suisse or Dunod, Paris, (2 volumes), 1983.

[FLI 11] FLIELLER D., Commande d‘actionneurs synchrones en défaut, Hermès, Paris, forthcoming.

[GRE 97] GRELLET G., CLERC G., Actionneurs électriques – Principes – Modèles – Commande, Eyrolles, Paris, 1997.

[GRE 93] GRENIER D., LOIUS J.-P., “Modelling for control of a non-sinewave permanent magnet synchronous drive by an extension of Parks transfromation”, Proc. 4^th IMACS TC1, Montreal, Canada, 7-9 July 1993.

[GRE 95] GRENIER D., LOIUS J.-P., “Modelling for control of a non-sinewave permanent magnet synchronous drives by an extension of Parks transformation”, Mathematics and Computers in Simulation, p. 445-452, 1995.

[GRE 97] GRENIER D., DESSAINT L.A., Akhrif O., Bonnassieux Y., Le Pioufle B., “Experimental nonlinear torque control of a permanent magnet synchronous motor using saliency”, IEEE Transactions on Industrial Electronics, vol. 44, no. 5, p. 680-687, 1997.

[GRE 98] GRENIER D., YALA S., LOIUS J.-P., “Définition d‘extensions de la transformation de Park pour la commande de moteurs synchrones à aimants permanents montés en surface”, European Physics Journal AP 1, p. 233-246, 1998.

[GRE 01] GRENIER D., LABRIQUE F., BUYSE H., MATAGNE E., Electromécanique, convertisseurs d‘énergie et actionneurs, Dunod, Paris, 2001.

[GRE 04] GRENIER D., STURTZER G., FLIELLER D., LOIUS J.-P., “Extension de la transformée de Park aux moteurs synchrones à distribution de champ non sinusöidale”, in Louis J.-P. (ed.), Modèles pour la commande des actionneurs électriques (EGEM treatise, Electrical Engineering Series), Hermès, Paris, 2004.

[HAU 97] HAUTIER J.-P., CARON J.-P., Systèmes automatiques 2, Commande des processus, Ellipses, Paris, 1997.

[HUS 03] HUSSON R. (ed.), Méthode de commande des machines électriques, Hermès, Paris, 2003.

[LAB 95] LABRIQUE F., SÉGUIER G., BAUSIÈRE R., Les convertisseurs de l‘électronique de puissance 4, La conversion continu-alternatif, 2^nd edition, Tec & Doc, Paris, 1995.

[LAB 04] LABRIQUE F., LOIUS J.-P., “Modélisation des onduleurs de tension en vue de leur commande en M.L.I.”, in Louis J.-P. (ed.), Modèles pour la commande des actionneurs électriques (EGEM treatise, Electrical Engineering Series), Hermès, Paris, 2004.

[LAB 11] LABRIQUE F., LOIUS J.-P., “Carrier-based pulse width modulation for two-level three-phase voltage inverters”, in Monmasson E. (ed.), Power Electronic Converters: PWM Strategies and Current Control Techniques, ISTE, London and John Wiley & Sons, New York, 2011.

[LAJ 95] LAJOIE-MAZENC M., VIAROUGE P., “Alimentation des Machines synchrones”, traité de génie électriques, D3630 and D 3631, Techniques de l‘Ingénieur, 1991.

[LEO 90] LEONHARD W., Control of electrical machines, Springer-Verlag, Berlin, Germany, 1990.

[LED 09] LE DÆUFF R., EL-HADI ZAÏM M., Machines électriques tournantes, de la modélisation matricielles à la mise en œuvre, Hermès, Paris, 2009.

[LEP 90] LE PIOUFLE B., GEORGIOU G., LOIUS J.-P., “Application des Commandes non- linéaires pour la Régulation en vitesse ou en Position de la Machine synchrone self- controlée”, Revue de Physique appliquée, no. 25, p. 517-526, June 1990.

[LES 81] LESENNE J., NOTELET F., Séguier G., Introduction à l‘électrotechnique approfondie, Tec & Doc, Paris, 1981.

[LAC 94] LACROUX G., Les actionneurs électriques pour la robotique et les asservissements, Tec & Doc, Paris, 1994.

[LOR 03] LORON L. (ed.), Commande des systèmes électriques: perspectives technologiques (EGEM treatise, Electrical Engineering Series), Hermès, Paris, 2003.

[LOU 99] LOIUS J.-P., BERGMANN C., “Commande Numérique des Machines synchrones”, Techniques de l‘Ingénieur, D 3644, Paris, May 1999.

[LOU 04a] LOIUS J.-P. (ed.), Modélisation des machines électriques en vue de leur commande, Concepts généraux (EGEM treatise, Electrical Engineering series), Hermès, Paris, 2004.

[LOU 04b] LOIUS J.-P. (ed.), Modèles pour la commande des actionneurs électriques (EGEM treatise, Electrical Engineering Series), Hermès, Paris, 2004.

[LOU 04c] LOIUS J.-P., FELD G., MOREAU S., “Modélisation physique des machines à courant alternatif”, in Louis J.-P. (ed.), Modélisation des machines électriques en vue de leur commande, Concepts généraux (EGEM treatise, Electrical Engineering series), Hermès, Paris, 2004.

[LOU 04d] LOIUS J.-P., FELD G., MONMASSON E., “Modélisation des machines à courant alternatif par les phaseurs”, in Louis J.-P. (ed.), Modélisation des machines électriques en vue de leur commande, Concepts généraux (EGEM treatise, Electrical Engineering series), Hermès, Paris, 2004.

[LOU 09] LOIUS J.-P., HOANG E., FELD G., Actionneurs à courant continu et synchrone self- controlé – choix, dimensionnements, commande d‘axe - un problème de synthèse, Hermès, 2009.

[MAR 92] MARCHAND C., RAZEK A., “Electromagnetic modelling to optimize low speed and position control in servo motor”, International Workshop on Electric and Magnetic Fields from NumE.al Models to Industrial Applications, Liège, Belgium, 28-30 September 1992.

[MAT 04] MATAGNE E., DA SILVA GARRIDO M., “Conversion électromécanique d‘énergie: du phénomène physique à la modélisation dynamique”, in Louis J.-P. (ed.), Modélisation des machines électriques en vue de leur commande, Concepts généraux (EGEM treatise, Electrical Engineering series), Hermès, Paris, 2004.

[MIL 89] MILLER T.J., Brushless Permanent Magnet and Reluctance Motor Drives, Oxford University Press, New York, United States, 1989.

[MON 04] MONTEIRO J.R.B., Transformacao da Nao Senoidal para Maquinas Sıncronas com Ima Permanente no Rotor, Electrical Engineering PhD Thesis, Sao Carlos Engineering School, Sao Paulo Univeristy, Brazil, 2004.

[MON 11] MONMASSON E. (ed.), Power Electronic Converters: PWM Strategies and Current Control Techniques, ISTE, London and John Wiley & Sons, New York, 2011.

[MUL 06] MULTON B., BEN AHMED H., RUELLAN M., ROBIN G., “Comparaison du couple massique de diverses architectures de machines tournantes synchrones à aimants = Comparison of torque to mass ratio of several architectures of permanent magnet synchronous machines”, Revue de l‘électricité et de l‘électronique, no. 8, p. 85-93, 2006.

[PAR 28a] PARK R. H., “Definition of an ideal synchronous machine and formula for the armature flux linkages”, General Electric Revue, p. 332-334, June 1928.

[PAR 28b] PARK R. H., ROBERTSON B. L., “The reactance of synchronous machines”, A.I.E.E. Quarterly Trans., vol. 47, no. 2, p. 514, April 1928.

[PAR 29] PARK R. H., “Two-reaction theory of synchronous machines, generalized method of analysis, part 1”, Transactions A.I.E.E. (American Institute of Electrical Engineers), vol. 48, no.3, p. 716-727, July 1929.

[PAR 33] PARK R. H., “Two-reaction theory of synchronous machines, generalized method of analysis, part 2”, Transactions A.I.E.E. (American Institute of Electrical Engineers), vol. 52, no. 2, p. 716-730, July 1933.

[ROB 07] ROBYNS B., FRANÇOIS B., DEGOBERT P., HAUTIER J.-P., Commande vectorielle de la machine asynchrone – Désensibilisation et optimisation par la logique floue, TECHNIP, Paris, 2007.

[SAR 04] SARGOS F.-M., MEIBODY-TABAR F., “Modèles dynamiques des machines synchrones”, Chapter 2, p. 55-148, in Louis J.-P. (ed.), Modèles pour la commande des actionneurs électriques (EGEM treatise, Electrical Engineering Series), Hermès, Paris, 2004.

[SEM 04] SEMAIL E., LOIUS J.-P., FELD G., “Propriétés vectorielles des systèmes électriques triphasés”, in Louis J.-P. (ed.), Modélisation des machines électriques en vue de leur commande, Concepts généraux (EGEM treatise, Electrical Engineering series), Hermès, Paris, 2004.

[STU 00b] STURTZER G., SMIGEL E., Modélisation et commande des moteurs triphasés – Commande vectorielle des moteurs synchrones – Commande numérique par contrôleur DSP, Ellipses, Paris, 2000.

[STU 01] STURTZER G., Modèle inverse et réduction de l‘ondulation de couple pour machines synchrones déduits des courbes isocouples. Extension de la transformation de Park pour moteurs synchrones à pôles saillants non sinusoïdaux et saturés, PhD Thesis, ENS Cachan, 28 November 2001.

[TOU 11] TOUNZI A., “Alimentation et commande des actionneurs à réluctance”, in J.-P Louis (ed.), Commande d‘actionneurs synchrones spéciaux, Hermès, forthcoming.

[VAS 90] VAS P., Vector Control of AC Machines, Clarendon Press Oxford, United States, 1990.

[WIL 05] WILDI T., SYBILLE G., Electrotechnique, 4^th edition, Presses de l‘université Laval, Quebec, Canada, 2005.

[XIA 89] XIAQJUN Z. L., VERGHESE G. C., LANG J. H., ONDER M. K., “Generalizing the Blondel-Park transformation of electrical machines: necessary and sufficient conditions”, IEEE Trans Transactions on Circuits and Systems, vol. 36, no.8, p. 1058-1067, 1989.

---

1 Chapter written by Jean-Paul LOIUS, Damien FLIELLER, Ngac Ky NGUYEN and Guy STURTZER.

1. By convention, because we consider that the typical functioning is the motor functioning, we call the induced voltages: “counter electromotive forces” [WIL 05].

2. α is a classical notation, that we will not confuse with the two-phase component, seen in section 1.5.2.